3.2155 \(\int \left (a+b \sqrt{x}\right )^{10} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b \sqrt{x}\right )^{12}}{6 b^2}-\frac{2 a \left (a+b \sqrt{x}\right )^{11}}{11 b^2} \]

[Out]

(-2*a*(a + b*Sqrt[x])^11)/(11*b^2) + (a + b*Sqrt[x])^12/(6*b^2)

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Rubi [A]  time = 0.0508927, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (a+b \sqrt{x}\right )^{12}}{6 b^2}-\frac{2 a \left (a+b \sqrt{x}\right )^{11}}{11 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10,x]

[Out]

(-2*a*(a + b*Sqrt[x])^11)/(11*b^2) + (a + b*Sqrt[x])^12/(6*b^2)

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Rubi in Sympy [A]  time = 13.3855, size = 32, normalized size = 0.84 \[ - \frac{2 a \left (a + b \sqrt{x}\right )^{11}}{11 b^{2}} + \frac{\left (a + b \sqrt{x}\right )^{12}}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**10,x)

[Out]

-2*a*(a + b*sqrt(x))**11/(11*b**2) + (a + b*sqrt(x))**12/(6*b**2)

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Mathematica [B]  time = 0.0191046, size = 131, normalized size = 3.45 \[ a^{10} x+\frac{20}{3} a^9 b x^{3/2}+\frac{45}{2} a^8 b^2 x^2+48 a^7 b^3 x^{5/2}+70 a^6 b^4 x^3+72 a^5 b^5 x^{7/2}+\frac{105}{2} a^4 b^6 x^4+\frac{80}{3} a^3 b^7 x^{9/2}+9 a^2 b^8 x^5+\frac{20}{11} a b^9 x^{11/2}+\frac{b^{10} x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10,x]

[Out]

a^10*x + (20*a^9*b*x^(3/2))/3 + (45*a^8*b^2*x^2)/2 + 48*a^7*b^3*x^(5/2) + 70*a^6
*b^4*x^3 + 72*a^5*b^5*x^(7/2) + (105*a^4*b^6*x^4)/2 + (80*a^3*b^7*x^(9/2))/3 + 9
*a^2*b^8*x^5 + (20*a*b^9*x^(11/2))/11 + (b^10*x^6)/6

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Maple [B]  time = 0.003, size = 110, normalized size = 2.9 \[{\frac{{x}^{6}{b}^{10}}{6}}+{\frac{20\,a{b}^{9}}{11}{x}^{{\frac{11}{2}}}}+9\,{x}^{5}{a}^{2}{b}^{8}+{\frac{80\,{a}^{3}{b}^{7}}{3}{x}^{{\frac{9}{2}}}}+{\frac{105\,{x}^{4}{a}^{4}{b}^{6}}{2}}+72\,{x}^{7/2}{a}^{5}{b}^{5}+70\,{x}^{3}{a}^{6}{b}^{4}+48\,{x}^{5/2}{a}^{7}{b}^{3}+{\frac{45\,{x}^{2}{a}^{8}{b}^{2}}{2}}+{\frac{20\,{a}^{9}b}{3}{x}^{{\frac{3}{2}}}}+x{a}^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10,x)

[Out]

1/6*x^6*b^10+20/11*x^(11/2)*a*b^9+9*x^5*a^2*b^8+80/3*x^(9/2)*a^3*b^7+105/2*x^4*a
^4*b^6+72*x^(7/2)*a^5*b^5+70*x^3*a^6*b^4+48*x^(5/2)*a^7*b^3+45/2*x^2*a^8*b^2+20/
3*x^(3/2)*a^9*b+x*a^10

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Maxima [A]  time = 1.43887, size = 41, normalized size = 1.08 \[ \frac{{\left (b \sqrt{x} + a\right )}^{12}}{6 \, b^{2}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a}{11 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10,x, algorithm="maxima")

[Out]

1/6*(b*sqrt(x) + a)^12/b^2 - 2/11*(b*sqrt(x) + a)^11*a/b^2

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Fricas [A]  time = 0.230588, size = 153, normalized size = 4.03 \[ \frac{1}{6} \, b^{10} x^{6} + 9 \, a^{2} b^{8} x^{5} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 70 \, a^{6} b^{4} x^{3} + \frac{45}{2} \, a^{8} b^{2} x^{2} + a^{10} x + \frac{4}{33} \,{\left (15 \, a b^{9} x^{5} + 220 \, a^{3} b^{7} x^{4} + 594 \, a^{5} b^{5} x^{3} + 396 \, a^{7} b^{3} x^{2} + 55 \, a^{9} b x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10,x, algorithm="fricas")

[Out]

1/6*b^10*x^6 + 9*a^2*b^8*x^5 + 105/2*a^4*b^6*x^4 + 70*a^6*b^4*x^3 + 45/2*a^8*b^2
*x^2 + a^10*x + 4/33*(15*a*b^9*x^5 + 220*a^3*b^7*x^4 + 594*a^5*b^5*x^3 + 396*a^7
*b^3*x^2 + 55*a^9*b*x)*sqrt(x)

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Sympy [A]  time = 4.78175, size = 133, normalized size = 3.5 \[ a^{10} x + \frac{20 a^{9} b x^{\frac{3}{2}}}{3} + \frac{45 a^{8} b^{2} x^{2}}{2} + 48 a^{7} b^{3} x^{\frac{5}{2}} + 70 a^{6} b^{4} x^{3} + 72 a^{5} b^{5} x^{\frac{7}{2}} + \frac{105 a^{4} b^{6} x^{4}}{2} + \frac{80 a^{3} b^{7} x^{\frac{9}{2}}}{3} + 9 a^{2} b^{8} x^{5} + \frac{20 a b^{9} x^{\frac{11}{2}}}{11} + \frac{b^{10} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**10,x)

[Out]

a**10*x + 20*a**9*b*x**(3/2)/3 + 45*a**8*b**2*x**2/2 + 48*a**7*b**3*x**(5/2) + 7
0*a**6*b**4*x**3 + 72*a**5*b**5*x**(7/2) + 105*a**4*b**6*x**4/2 + 80*a**3*b**7*x
**(9/2)/3 + 9*a**2*b**8*x**5 + 20*a*b**9*x**(11/2)/11 + b**10*x**6/6

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GIAC/XCAS [A]  time = 0.21836, size = 147, normalized size = 3.87 \[ \frac{1}{6} \, b^{10} x^{6} + \frac{20}{11} \, a b^{9} x^{\frac{11}{2}} + 9 \, a^{2} b^{8} x^{5} + \frac{80}{3} \, a^{3} b^{7} x^{\frac{9}{2}} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 72 \, a^{5} b^{5} x^{\frac{7}{2}} + 70 \, a^{6} b^{4} x^{3} + 48 \, a^{7} b^{3} x^{\frac{5}{2}} + \frac{45}{2} \, a^{8} b^{2} x^{2} + \frac{20}{3} \, a^{9} b x^{\frac{3}{2}} + a^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10,x, algorithm="giac")

[Out]

1/6*b^10*x^6 + 20/11*a*b^9*x^(11/2) + 9*a^2*b^8*x^5 + 80/3*a^3*b^7*x^(9/2) + 105
/2*a^4*b^6*x^4 + 72*a^5*b^5*x^(7/2) + 70*a^6*b^4*x^3 + 48*a^7*b^3*x^(5/2) + 45/2
*a^8*b^2*x^2 + 20/3*a^9*b*x^(3/2) + a^10*x